Exploring Areas

Kim Seay

EMAT 6680



To begin this problem, we need a rectangle. Next, choose a point anywhere on one of the sides of the rectangle. Form a triangle that has two shared vertices and one shared side with the rectangle. Click here for a GSP sketch that will allow you to choose any three points and create a figure such as this.

The problem asks students to compare the area of the triangle with that of the rectangle.

In this case the area of our triangle is half the area of the rectangle, or we could say the ratio of areas from rectangle to triangle is 2:1. We can click on point F at the top of the triangle and drag it along the side of the rectangle to investigate if this will change the ratio of the areas. Click here to try this.

As you can see, the ratio will not change regardless of the location of point p. In fact, this ratio will be the same given any rectangle and a triangle that shares two of its vertices and one side. This makes sense when we look at the formulas for finding the area.

Area of a rectangle = length x height = l x h

Area of a triangle = 1/2 base x height = 1/2bh

In this figure, height of the rectangle and the triangle will always be the same. The base of the triangle is the same as the length of the rectangle. To find the area of the triangle, we simply take the area of the rectangle and multiply by 1/2.


I chose this application, because it is very applicable to 6th grade math. This is a good example of how GSP can be used in the middle school classroom.